Average Error: 58.4 → 27.2
Time: 2.0m
Precision: 64
Internal Precision: 7488
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right) \cdot \left(\frac{c0}{2 \cdot w}\right) + \left(\left(\left(e^{\log \left(\sqrt[3]{\frac{c0}{w \cdot h}}\right)} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\sqrt[3]{\frac{c0}{w \cdot h}} \cdot \sqrt[3]{\frac{c0}{w \cdot h}}\right)\right) \cdot \frac{c0}{2 \cdot w}\right))_* \le 5.745113171430627 \cdot 10^{+276}:\\ \;\;\;\;\sqrt[3]{(\left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right) \cdot \left(\frac{c0}{2 \cdot w}\right) + \left(\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right)\right))_*} \cdot \left(\sqrt[3]{(\left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right) \cdot \left(\frac{c0}{2 \cdot w}\right) + \left(\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right)\right))_*} \cdot \sqrt[3]{(\left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right) \cdot \left(\frac{c0}{2 \cdot w}\right) + \left(\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right)\right))_*}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Derivation

  1. Split input into 2 regimes

  2. if (fma (sqrt (fma (* (/ c0 (* h w)) (* (/ d D) (/ d D))) (* (/ c0 (* h w)) (* (/ d D) (/ d D))) (* M (- M)))) (/ c0 (* w 2)) (* (/ c0 (* w 2)) (* (* (cbrt (/ c0 (* h w))) (cbrt (/ c0 (* h w)))) (* (exp (log (cbrt (/ c0 (* h w))))) (* (/ d D) (/ d D)))))) < 5.745113171430627e+276

    1. Initial program 46.3

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

    2. Initial simplification22.6

      \[\leadsto (\left(\sqrt{(\left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(M \cdot \left(-M\right)\right))_*}\right) \cdot \left(\frac{c0}{w \cdot 2}\right) + \left(\frac{c0}{w \cdot 2} \cdot \left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right))_*\]
    3. Using strategy rm

    4. Applied add-cube-cbrt22.9

      \[\leadsto \color{blue}{\left(\sqrt[3]{(\left(\sqrt{(\left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(M \cdot \left(-M\right)\right))_*}\right) \cdot \left(\frac{c0}{w \cdot 2}\right) + \left(\frac{c0}{w \cdot 2} \cdot \left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right))_*} \cdot \sqrt[3]{(\left(\sqrt{(\left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(M \cdot \left(-M\right)\right))_*}\right) \cdot \left(\frac{c0}{w \cdot 2}\right) + \left(\frac{c0}{w \cdot 2} \cdot \left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right))_*}\right) \cdot \sqrt[3]{(\left(\sqrt{(\left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(M \cdot \left(-M\right)\right))_*}\right) \cdot \left(\frac{c0}{w \cdot 2}\right) + \left(\frac{c0}{w \cdot 2} \cdot \left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right))_*}}\]

    if 5.745113171430627e+276 < (fma (sqrt (fma (* (/ c0 (* h w)) (* (/ d D) (/ d D))) (* (/ c0 (* h w)) (* (/ d D) (/ d D))) (* M (- M)))) (/ c0 (* w 2)) (* (/ c0 (* w 2)) (* (* (cbrt (/ c0 (* h w))) (cbrt (/ c0 (* h w)))) (* (exp (log (cbrt (/ c0 (* h w))))) (* (/ d D) (/ d D))))))

    1. Initial program 60.4

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

    2. Initial simplification61.4

      \[\leadsto (\left(\sqrt{(\left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) + \left(M \cdot \left(-M\right)\right))_*}\right) \cdot \left(\frac{c0}{w \cdot 2}\right) + \left(\frac{c0}{w \cdot 2} \cdot \left(\frac{c0}{h \cdot w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\right))_*\]

    3. Taylor expanded around inf 28.0

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification27.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;(\left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right) \cdot \left(\frac{c0}{2 \cdot w}\right) + \left(\left(\left(e^{\log \left(\sqrt[3]{\frac{c0}{w \cdot h}}\right)} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\sqrt[3]{\frac{c0}{w \cdot h}} \cdot \sqrt[3]{\frac{c0}{w \cdot h}}\right)\right) \cdot \frac{c0}{2 \cdot w}\right))_* \le 5.745113171430627 \cdot 10^{+276}:\\ \;\;\;\;\sqrt[3]{(\left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right) \cdot \left(\frac{c0}{2 \cdot w}\right) + \left(\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right)\right))_*} \cdot \left(\sqrt[3]{(\left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right) \cdot \left(\frac{c0}{2 \cdot w}\right) + \left(\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right)\right))_*} \cdot \sqrt[3]{(\left(\sqrt{(\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right) + \left(\left(-M\right) \cdot M\right))_*}\right) \cdot \left(\frac{c0}{2 \cdot w}\right) + \left(\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w \cdot h}\right)\right))_*}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))